# Solve the following logarithmic equation Find the domain of fog, `f(x)=5/(x-3)` , `g(x)=1/x` .That is 5 over x-3 and 1 over x

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### 2 Answers

The domain of the composition of functions is the intersection of the domains of the argument of the composition with the final composition function.

This means that we can look at where the argument function is "bad" to find the spots where the composition is also going to be bad.

Since `g(x)=1/x` has domain `{x in R|x ne 0}` , and the composition function is found by some algebra:

`f circ g(x)`

`=f(g(x))`

`=f(1/x)`

`=5/{1/x-3}`

`=5/{{1-3x}/x}`

`={5x}/{1-3x}`

which also has a vertical asymptote at `x=1/3` .

**This means that the domain of the composition is `{x in R|x ne 0, x ne 1/3}` **

`f(x)= 5/(x-3)`

`g(x)=1/x`

To solve for the composite function *f o g*, replace the x in f(x) with 1/x.

`f@g=f(g(x)) = f(1/x) = 5/(1/x-3) = 5/[(1-3x)/x]`

`f@g= (5x)/(1-3x)`

Since f o g is a rational function, we need to take note that we cannot have a zero denominator. So the values of x that make the denominator equal to zero is:

1-3x = 0

-3x = -1

x = 1/3

So, x should not be equal to 1/3.

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**Hence, `f@g= (5x)/(1-3x)` . Its domain is all real numbers except `1/3` .**